The case of the terrible tires

Doug was a good driver. He was on a dry asphalt highway with a curve to the left. It was surprising that the accident had such a tragic outcome. Doug said later that he was gradually braking his '95 Toyota T100 pickup as a deer approached the road. As the deer crossed the road, he increased braking and lost control of the pickup as the rear end swerved around to the right, taking him onto a sloped shoulder. The pickup rolled, leaving Doug, who was not wearing a seat belt, a paraplegic.

Investigation at the scene revealed two skidmarks left by the rear tires. All evidence indicated that the front tires were not skidding. Vehicle inspection indicated that the front brakes were operational, implying that the front tires were braking but not skidding, even though the rear tires were skidding. Why?

When a vehicle is rolling at constant speed with no braking, the friction between the tires and the road is the "rolling resistance," and there is no slipping. As braking begins, the braking torque (TF = front, TR = rear) is opposed by the friction torque produced about the center of the wheels, as shown in the sketch. The friction forces are opposite to the motion, decelerating the vehicle. Some stretching of the tread occurs due to the increasing friction, and there is partial slipping. Skidding (100% slip) occurs when the braking torque exceeds the torque produced by the maximum friction force, µ× N. N is the normal force resulting from the road pressure, and µ is the coefficient of friction. The maximum resisting friction torque is µ× N × r, where r is the rolling radius of the tire. There is little directional control with a tire that is skidding, since the available friction is all utilized in opposing the relative motion, and little is available for steering or lateral stability.

Examination of Doug's pickup revealed that the tires on the rear were not the same size as the front tires. Doug had put new tires on his wife's Toyota Avalon, and used the two best ones to replace the worn rear tires on his T100. The original T100 tires were P215/75 R15S and those on the Avalon were P205/65R15 92H. The tire numbers appear pretty close, which might make you think it is okay to mix them. However, the aspect ratio (the number following the forward slash) representing the ratio of the height of the cross section to the width, is significantly different for the two tires. This means that the radius (rF) of the original T100 tires is 13.8", while the radius (rR) of the Avalon tires is only 12.7". If braking system design is such that the friction force is the same on the front and rear axles, the maximum braking torque that can be opposed by friction is T FMAX = µ× N × rF (front) and T RMAX = µ× N x rR (rear). Note that T RMAX = (rR/ rF) x T FMAX = 0.92 × T FMAX, indicating that the rear would begin to slip first. Since Doug applied the braking gradually, the rear axle began to skid while the front axle was still rolling. This is a very unstable situation since there is no directional stability of the rear wheels. The centrifugal force due to the slight curve in the road pushed the rear wheels to the right, rotating the pickup about the front wheels.

This case illustrates one reason why it is important to be sure you have a matched set of tires on your car. Some tire companies will not sell you fewer than four snow tires at a time, because of the serious consequences of having mismatched tires. Tires of the same size can have different radii for several reasons, such as wear, under-inflation, overloading, etc. These situations can also lead to serious stability problems and should be avoided.

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